Today: ASV 4.3 Next: ASV 4ynemish/180a/180alecture16afterClass.pdf · 2019-11-02 ·...
Transcript of Today: ASV 4.3 Next: ASV 4ynemish/180a/180alecture16afterClass.pdf · 2019-11-02 ·...
MATH180A: Introduction to Probability
www.math.ucsd.edu/~ynemish/180a
This week:
homework #4 (due Friday, Nov 1, 11:59pm)
Today: ASV 4.3
Next: ASV 4.4
-
( LT for the binomial
÷÷ïï÷ü÷ü"iRate of thumb :
For Are average ¥ npa - p ) > 10
p(q¥" ± ¥ -p±bY)#(b) -Pla)
Œ"NË:bËË?p;Ymmenaray
confidenceintervals.NO#tioh
Independent trials ,success rate p (unknown )
Sn = member of successes after n trials
Sn - Bon , p) . ( E. g. : ( biased ?) coin tossed n timesHo
LLN : SI → p , nao (in probabitity )If in is big , ¥ is close to p .
randommember
variable ← observable,estimant of p
Usually ,we don't know p , but we can get a
réalisation of § (flipping coin) for finit n .
M .
What can we soy about p ?
ConfidenceintervalssetupcLT : internal C-a. a) pr §← estimateur
pfqfmtT-ep-peaff-T.pt ) = 29cal - t
Pf tp -pk af¥ ) = Ca) - t = :XÛ confidence level
Questions : fixed (find a)1) For fixed n ,
find Est .
-
PUP -pkE)et |2) For fixed E ,find ns.t .
PUÊ - pk E)Et l
Confidenceintervalsp( pt [ pr - e , pâte 3) ± y[ § - E , pr +E ) - y - confidence internal for p
§ is r - v . ⇒ internal is random .
= EnTata some réalisation of pr (a member) , say Ê*
[ ÎE , Ê# E) - y - confidence internal of p
Fat → est morte of p
ConfidenceintervalscomputationsoWhat with unknown pin formula ?
Pl In -pkavpa.pt.
P f) = 2 (a) - t = :p
N
Par - ne ÷}? :{fÏÏThe j - confidence Internal can
be taken as
[ § - E , ptite ] with• E >Ça forfixed n
• Tn > zaz for fixede
and 2 (a) - II y 20pct
cahidenceintervals.Exampfixednjfh.pecoin 10000 times .
Num best of Heads is 5370.
Compute a 99% - confidence Internal for p=P( Heads)M = 10000
, §* = 5370 = 0-537
201/2 - VIOOOT - E) - I I 0.99 0
(2- 100 - E) ? 0.995
2- IDO - E 22.58
{ I 2.58-200 ⇒ EZO - 0129
99% - confidence internat for p o ois given by [0.537 - 0.0129 , 0.537+0-0129]
Confidence internats . Example 2 ixed accuracy)
Flip a potentially biased coin . Hou mang times
should we repeat the expériment to be able to
compute a 95% - confidence internal for p = PCHeads)µ
of length 0.01 ?
j' = 0.95 ,E = 0.005
210fr -t - e) - t
= 2 (2- 0.005 - Fn ) - I I 0.95( O - 01 - Fn ) Z O - 975
0.01 . Fn ? 1.96 ⇒ nz 496)?- O
g- 95%- conf . Internal
I Repeat 1962 times, compute Ê* , [ pt - o.oot.pe +0.005)
for P
Confidenceintervalspollingwithoutgoinginto détails (see Example a -m)- Remark 4. 16
Part of population (unknown p) préfets product A- /supports candidate Bl. . .
We interview n individual s and K of them ogivepositive censurer (about product Kandi date) .
What can we soy about p ?
Fix confidence level j (often y = 0.95 )KAgain , pq is the estimât of p ,
[ ¥ - E , ¥ RE) gives 2 Kern ) - t - confidence infernal
⇒ 2 (« fn ) -pzz→ fixe d sample size→ fixedaccuracy
Confidentiellement rock vs rap )You ask 400 random1g chose n people living in SDif they prefer rock or rap . 230 reply that
they prefere rock music . Give a 99% confidence
internal for the part of the population that préfets rock .
• n = 400 , y = 0.99 , § = 24¥ = 0.575
2f(2E f407 ) - I I 0.99 ⇒ (40 E) I O - 995
=) 40 E I 2.58 E I = O.O 645
575% wittr[ 0.575 - O -0645
,0.575+0.0645 ) ← margin error 6.45%
•